Thursday, October 15, 2009

The endo, short for end-over or end-over-end, is a type of pitch over crash where the cyclist goes over the handlebars, the weight offset of which causes an inertial moment to act about the front wheel resulting in rear portion of the bicycle to flip in the air above and behind him.

Usually, the cyclist, as a sudden reflex action, yanks out their hands or legs at some point to cushion the impending fall and ends up letting go of handlebar control. Meanwhile, the bicycle is bound to fall either sideways, due to its motion about the steering axis or right on top of the cyclist. In the latter scenario, the saddle or even the rear wheel itself could land on the cyclist's body.

Again, an endo is a crash that could cause injury. It is not a bicycle trick. That one has another name. Its called a 'stoppie' or a 'wheelie'. An endo is caused due to strong front wheel braking or when the bicycle hits a curb, a structure more rigid than the wheel itself. Endos may also occur if the front wheel is loose, i.e, if it is not secured properly to the fork dropouts by the quick release skewers.

In this post, we will cover the "endo parameter", study the relationship between braking force and endo parameter on level ground, outline some common reasons for endos, check out a video analysis of an endo and finally study the relationship between gradient of the road and endo parameter through a literature source.

ENDO PARAMETER

Braking a bicycle naturally upsets equilibrium and transfers weight to the front wheel. With a stark increase in the overall braking force, the load on the rear wheel approaches zero, after which the rear wheel will start to lift off the ground. Hard braking may stop the bicycle but Newton's first law reigns supreme as the cyclist's body continues in motion in the headed direction. This rider motion has some momentum. If not self-controlled, the rider will flip over the handlebars and the bicycle will pitch-over as well. What results is the endo.

It turns out that while outrageous situations cannot be helped, some factor of safety from bicycle design and rider positioning skill can provide for a cushion against pitch-over tendency in the above mentioned situations.

I'll call my main parameter of interest the pitch-over parameter (or endo parameter for lack of a better word) - A/H - as can be seen in the diagram below :

Fig 1 : Free Body diagram of a bicycle-rider system just at pitch-over. Now you may be able to appreciate from geometry and c.o.g as to why recumbents and tandems are stable in pitch-over. O is the point signifying the contact point of the front wheel with the ground.

The endo parameter, A/H (a ratio), in the combined bicycle-rider system should be large enough to avoid front pitch-over. Obviously the vertical height, H, of the center of gravity (c.o.g) and the location of the c.o.g aft of the front wheel, A, are going to vary with variation in rider's weight, height and sitting position.

This highlights why its important to get a proper bike fit for the type of bicycle you wish to ride. Its not just a question about comfort. Its also a question about safety. Enlarged riders who overwhelm miniature bikes not made for their size will quickly find out what they're doing wrong. All they have to do is hit the front brakes hard and they're right on target to be turned into human projectiles.

Looking at the free body diagram above, we can deduce that the system is in static equilibrium about the front wheel contact point O if the sum of the moments due to all forces about that point is zero. In other words, rotation is just initiated at :

Fig 2 : Endo parameter relationship to braking force. This also gives us an expression for the braking force at the point of the pitch-over.

From this simple relation for level ground, we see that endo parameter is equal to the braking force as a percentage of total weight at just about the initiation of the endo. Braking force is a function of the co-efficient of friction at the tire-road interface.

Before the initiation of pitch-over, the braking force-weight ratio is lesser than the endo parameter. Well after the pitch-over has been initiated, the endo parameter falls lesser than the braking force-weight ratio.

We can now infer that making A/H larger is better for safety. Otherwise, a lesser braking force relative to total weight will be sufficient to initiate pitch-over. How? Simply because the braking force-weight ratio catches up with the endo parameter sooner. Oops.

A/H can be fixed to be greater with good bicycle design and proper fit. It can also be superficially made larger by the cyclist while riding by positioning his body rearward (relative to bottom bracket) as the following picture shows :

Fig 3 : A cyclist ducks and shifts his c.o.g rearward to increase his endo parameter

People succumb to pitch-overs because of other factors too. They may not be skilled enough to increase the endo factor, A/H. They also may not be skilled enough to modulate and may tend to hitting the front brakes really hard without realizing that a front brake can cause more deceleration than a rear brake. For comparison, front brakes generate upto 0.5g's of retarding force whereas rear brakes produce a max of 0.1 or 0.2g's. Note that maximum deceleration is limited by the co-efficient of friction between tire and road and the normal load.

You can notice front braking power for some mountain bikes through a speed vs time graph.

Obviously, the graphs show that you can bring a bike to a stop faster using the front brakes alone than the rear brakes. Using both brakes is even better for reducing stopping distance even further.

SEQUENCE OF MOTIONS IN AN ENDO

Beck Forensics did an interesting little video analysis of an endo. The following image as well as the snippet below it is taken from a web sampler of their book Bicycle Collision Investigation. It shows the steps involved in an endo before the crash.

A mountain bicyclist traveling at about 22.5 mph (36.2 kph) applies only the front brake. Once the front wheel is nearly locked, the rear wheel starts to lift up. At about 0.20 seconds, the rider really has no chance to recover. At about 0.33 seconds, he releases the brake and prepares his right hand, and then his left hand for landing. The cones are shaped in 25 foot (7.6 m) intervals and the grade is about -2% (descent).(Courtesy : Beck Forensics)

RELATIONSHIP WITH PERCENTAGE GRADIENT OF GROUND

This section is a little more involved. It uses the same analysis techniques shown above to derive a relationship between the "endo parameter" and braking force-weight ratio with the percentage gradient of the ground. You will see that the chances of an endo are more likely on a descent.

The following literature is from one digest of IHPVA (2001), written by a retired engineer named Frederick Matteson. Click on the series of images to zoom the text. Alternatively, you can also read the paper here.

Here's an interesting endo story. I was out training in an early morning gray coastal fog in SoCal back in the 70s with a guy known for crashing into things. He was leading me up this 2-3% grade at high speed and I was focused solely on his rear wheel. I looked up briefly to see him hit the rear bumper of a gray car illegally parked in the bike lane. His bike broke at the head tube and he went flying off to the left. Next it was my turn and I hit the rear bumper squarely, breaking my front wheel. I did a flip over the entire car and landed on my back in front of it. My bike landed on the saddle some several feet ahead of me. I got up and the only damage to my person was a hole the size of my little finger in my wool Bianchi jersey. To this day I can barely believe what happened. Lesson: always look out ahead of the rider in front of you, no matter how red lined you are!

In physics, pushing an example to extremes often gives an insight into the solution. If instead of a 26 or 29er you had a 260er, two things would happen. First, your center of gravity would be under the front wheel axle, which is good. Foremost, with such a large wheel as you would begin to endo, the contact point of the front wheel to the ground would move forward in quite a sighificant way, thereby countering the lift effect. Actually, with a 26000er wheel, the endo would require the braking force to take you un 26000in into the air. No way that would happen at human speeds!

My conclusion is that a 29er would be slightly less prone to endoing, but probably not in a very significant way. To know how much exactly would require some calculations though.

This is a great discussion leading me to think that while we discuss 29'ers, keep in mind the importance of the role a set of tires can play, as well as fork suspension dampening. If you're going to test the endo tendency of a bike, wouldn't it be counterproductive riding in knobby tires on grassy, maybe even muddy trails with a suspension that can compress to a great deal?

Infact, when people say that 29'ers are less prone to endo-ing, how certain are they that their tires and suspension are not playing a big role in preventing the flip? Just a thought...

Here's a test one could design to test the theory that bigger wheels mean less endo. We need two bicycles, one 26'er and one 29'er. For both :

1. Use normal mountain biking tires on flat trail. No mud, no grassy patches that tires can sink into.

2. Eliminate suspension so that it doesn't skew observations.

3. Test endo tendency of bikes with similar weights and different weights.

4. Test endo tendency at both low and high speeds to see how aerodynamic drag affects what is observed.

5. Pay a rider well because its likely he'll be getting hurt in some fashion. :) No dummies allowed for the test. It just doesn't behave like a human being. Dummies don't ride bikes either.

Also, personal theories about 29'ers not endoing is just counterintuitive, based on the expression for flipover, which I have derived and shown in this post. The term for center of gravity occurs in the denominator and if we want to have a good chance of increasing the endo parameter, the expression says that center of gravity must be lowered and pushed as far back as possible in the event of hard braking. Increasing the center of gravity while keeping other parameters of bike design constant serves simply to lower the endo parameter limit, and sooner or later, our braking force is going to catch up with it, leading to an endo. Does anyone get me here, or am I rambling?

I don't have any scientific data to back it up, but my experience is the 29er is MORE likely to endo. The theory that the big front wheel would be harder to go over seems to make sense, but after flipping the bike time after time while descending a rough slope I have given up on the 29er. My best guess is that the 29" wheel in the rear hurts more than the 29" in the front helps.

Note that the A/H ratio is determined in part by how high the center of gravity is above the ground, not the front wheel's axle. I would argue further that the rider's forward momentum is eventually translated through a virtual lever that starts at the contact patch of the front tire and ends at the height of the grips on the handlebars. By using a larger diameter wheel/tire, the handlebar position, unless compensated for with shorter steer tubes, would be higher, thus resulting in a longer lever against which the same amount of force would be applied. The positive attribute of the niner wheel is possibly it's ability to better negotiate surface irregularities since the tire's outer edge creates less of an angle at impact with said irregularity